A fair eight-sided die is numbered 2, 2, 3, 3, 3, 4, 5 and 6. The die is rolled up to and including the roll on which the first 2 is obtained. Let X represent the number of times the die is rolled.

a. Find E(X).

b. Show that P(X ≥ 4) = 27/64.

c. The die is rolled up to and including the roll on which the first 2 is obtained on 20 occasions. Find, by use of a suitable approximation, the probability that X ≥ 4 on at least half of these 20 occasions.

d. Fully justify the approximation used in part c.